A non-commutative Mrowka’s Ψ-space

Piotr Koszmider

Institute of Mathematics of the Polish Academy of Sciences, Warsaw

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 1 / 8

Joint research with Saeed Ghasemi (IM PAN, Warsaw)

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 2 / 8

Ψ-spaces

DefinitionLet A be an almost disjoint family of subsets of N.We consider

ΨA = N ∪ {xA : A ∈ A}

with the following topology:elements of N are isolatedfor every A ∈ A all neighbourhoods of xA are of the form

UF (xA) = (A \ F ) ∪ {xA}.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 3 / 8

Ψ-spaces

DefinitionLet A be an almost disjoint family of subsets of N.

We consider

ΨA = N ∪ {xA : A ∈ A}

with the following topology:elements of N are isolatedfor every A ∈ A all neighbourhoods of xA are of the form

UF (xA) = (A \ F ) ∪ {xA}.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 3 / 8

Ψ-spaces

DefinitionLet A be an almost disjoint family of subsets of N.We consider

ΨA = N ∪ {xA : A ∈ A}

with the following topology:

elements of N are isolatedfor every A ∈ A all neighbourhoods of xA are of the form

UF (xA) = (A \ F ) ∪ {xA}.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 3 / 8

Ψ-spaces

DefinitionLet A be an almost disjoint family of subsets of N.We consider

ΨA = N ∪ {xA : A ∈ A}

with the following topology:elements of N are isolated

for every A ∈ A all neighbourhoods of xA are of the form

UF (xA) = (A \ F ) ∪ {xA}.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 3 / 8

Ψ-spaces

DefinitionLet A be an almost disjoint family of subsets of N.We consider

ΨA = N ∪ {xA : A ∈ A}

with the following topology:elements of N are isolatedfor every A ∈ A all neighbourhoods of xA are of the form

UF (xA) = (A \ F ) ∪ {xA}.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 3 / 8

Ψ-spaces

DefinitionLet A be an almost disjoint family of subsets of N.We consider

ΨA = N ∪ {xA : A ∈ A}

with the following topology:elements of N are isolatedfor every A ∈ A all neighbourhoods of xA are of the form

UF (xA) = (A \ F ) ∪ {xA}.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 3 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

and

σ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N)

such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

and

σ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

and

σ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

and

σ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

and

σ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

andσ[c0] is an essential ideal of B

the unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

andσ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e.,B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

andσ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

andσ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Mrowka’s Ψ-spaces

Theorem (Mrowka, 1977)There is an infinite almost disjoint family A ⊆ ℘(N) such β(ΨA) = α(ΨA).

Theorem (Mrowka, 1977 - an algebraic version)There an algebra B ⊆ `∞ which satisfies the following short exact sequence

0→ c0σ−→ B → c0(c)→ 0

andσ[c0] is an essential ideal of Bthe unitization of B is equal to the multiplier algebra of B, i.e., B =M(B).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 4 / 8

Main theorem

Theorem (S. Ghasemi, P. K.)There is a C∗-algebra A ⊆ B(`2) satisfying the following short exact sequence

0→ K(`2)σ−→ A → K(`2(c))→ 0,

such that

σ[K(`2)] is an essential ideal of Athe algebra of multipliersM(A) of A is equal to the unitization of A,

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 5 / 8

Main theorem

Theorem (S. Ghasemi, P. K.)There is a C∗-algebra A ⊆ B(`2) satisfying the following short exact sequence

0→ K(`2)σ−→ A → K(`2(c))→ 0,

such that

σ[K(`2)] is an essential ideal of Athe algebra of multipliersM(A) of A is equal to the unitization of A,

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 5 / 8

Main theorem

Theorem (S. Ghasemi, P. K.)There is a C∗-algebra A ⊆ B(`2) satisfying the following short exact sequence

0→ K(`2)σ−→ A → K(`2(c))→ 0,

such that

σ[K(`2)] is an essential ideal of Athe algebra of multipliersM(A) of A is equal to the unitization of A,

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 5 / 8

Main theorem

Theorem (S. Ghasemi, P. K.)There is a C∗-algebra A ⊆ B(`2) satisfying the following short exact sequence

0→ K(`2)σ−→ A → K(`2(c))→ 0,

such thatσ[K(`2)] is an essential ideal of A

the algebra of multipliersM(A) of A is equal to the unitization of A,

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 5 / 8

Main theorem

Theorem (S. Ghasemi, P. K.)There is a C∗-algebra A ⊆ B(`2) satisfying the following short exact sequence

0→ K(`2)σ−→ A → K(`2(c))→ 0,

such thatσ[K(`2)] is an essential ideal of Athe algebra of multipliersM(A) of A is equal to the unitization of A,

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 5 / 8

Main theorem

Theorem (S. Ghasemi, P. K.)There is a C∗-algebra A ⊆ B(`2) satisfying the following short exact sequence

0→ K(`2)σ−→ A → K(`2(c))→ 0,

such thatσ[K(`2)] is an essential ideal of Athe algebra of multipliersM(A) of A is equal to the unitization of A,

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 5 / 8

Main theorem

Theorem (S. Ghasemi, P. K.)There is a C∗-algebra A ⊆ B(`2) satisfying the following short exact sequence

0→ K(`2)σ−→ A → K(`2(c))→ 0,

such thatσ[K(`2)] is an essential ideal of Athe algebra of multipliersM(A) of A is equal to the unitization of A,

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 5 / 8

Matrix units and almost matrix units

FactA C∗-algebra A is isomorphic to the algebra K(`2(κ)) if and only ifit isgenerated by “matrix units” , that is nonzero elements (aβ,α : α, β ∈ κ)satisfying for each α, β, ξ, η < κ :

(aβ,α)∗ = aα,β ,aη,ξaβ,α = δξ,βaη,α.

DefinitionA sequence (aβ,α : α, β ∈ κ) of noncompact elements of B(`2) is called a“system of almost matrix units” if it satisfies for each α, β, ξ, η < κ:

(aβ,α)∗ =∗ aα,β ,aη,ξaβ,α =∗ δξ,βaη,α,

where a =∗ b means a− b ∈ K(`2).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 6 / 8

Matrix units and almost matrix units

FactA C∗-algebra A is isomorphic to the algebra K(`2(κ)) if and only if

it isgenerated by “matrix units” , that is nonzero elements (aβ,α : α, β ∈ κ)satisfying for each α, β, ξ, η < κ :

(aβ,α)∗ = aα,β ,aη,ξaβ,α = δξ,βaη,α.

DefinitionA sequence (aβ,α : α, β ∈ κ) of noncompact elements of B(`2) is called a“system of almost matrix units” if it satisfies for each α, β, ξ, η < κ:

(aβ,α)∗ =∗ aα,β ,aη,ξaβ,α =∗ δξ,βaη,α,

where a =∗ b means a− b ∈ K(`2).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 6 / 8

Matrix units and almost matrix units

FactA C∗-algebra A is isomorphic to the algebra K(`2(κ)) if and only ifit isgenerated by “matrix units” , that is nonzero elements (aβ,α : α, β ∈ κ)satisfying for each α, β, ξ, η < κ :

(aβ,α)∗ = aα,β ,aη,ξaβ,α = δξ,βaη,α.

DefinitionA sequence (aβ,α : α, β ∈ κ) of noncompact elements of B(`2) is called a“system of almost matrix units” if it satisfies for each α, β, ξ, η < κ:

(aβ,α)∗ =∗ aα,β ,aη,ξaβ,α =∗ δξ,βaη,α,

where a =∗ b means a− b ∈ K(`2).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 6 / 8

Matrix units and almost matrix units

FactA C∗-algebra A is isomorphic to the algebra K(`2(κ)) if and only ifit isgenerated by “matrix units” , that is nonzero elements (aβ,α : α, β ∈ κ)satisfying for each α, β, ξ, η < κ :

(aβ,α)∗ = aα,β ,

aη,ξaβ,α = δξ,βaη,α.

DefinitionA sequence (aβ,α : α, β ∈ κ) of noncompact elements of B(`2) is called a“system of almost matrix units” if it satisfies for each α, β, ξ, η < κ:

(aβ,α)∗ =∗ aα,β ,aη,ξaβ,α =∗ δξ,βaη,α,

where a =∗ b means a− b ∈ K(`2).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 6 / 8

Matrix units and almost matrix units

FactA C∗-algebra A is isomorphic to the algebra K(`2(κ)) if and only ifit isgenerated by “matrix units” , that is nonzero elements (aβ,α : α, β ∈ κ)satisfying for each α, β, ξ, η < κ :

(aβ,α)∗ = aα,β ,aη,ξaβ,α = δξ,βaη,α.

DefinitionA sequence (aβ,α : α, β ∈ κ) of noncompact elements of B(`2) is called a“system of almost matrix units” if it satisfies for each α, β, ξ, η < κ:

(aβ,α)∗ =∗ aα,β ,aη,ξaβ,α =∗ δξ,βaη,α,

where a =∗ b means a− b ∈ K(`2).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 6 / 8

Matrix units and almost matrix units

FactA C∗-algebra A is isomorphic to the algebra K(`2(κ)) if and only ifit isgenerated by “matrix units” , that is nonzero elements (aβ,α : α, β ∈ κ)satisfying for each α, β, ξ, η < κ :

(aβ,α)∗ = aα,β ,aη,ξaβ,α = δξ,βaη,α.

DefinitionA sequence (aβ,α : α, β ∈ κ) of noncompact elements of B(`2) is called a“system of almost matrix units” if it satisfies for each α, β, ξ, η < κ:

(aβ,α)∗ =∗ aα,β ,aη,ξaβ,α =∗ δξ,βaη,α,

where a =∗ b means a− b ∈ K(`2).

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 6 / 8

Example 1

For each ξ ∈ 2N we can associate a set Aξ = {s ∈ 2<N : s ⊆ ξ}.

FactLet X ⊆ N. Then for each λ ∈ {0,1} the sets {ξ ∈ 2N : Aξ ∩ X =∗ λAξ} areBorel.

For each pair (ξ, η) ∈ 2N × 2N we associate an operator on `2(2<N)

Tη,ξ(s) =

{eη|k if s = eξ|k for some k ∈ N0 otherwise

Lemma

Let R ∈ B(H) and U be a Borel subset of C, then the set

BRU = {(η, ξ) ∈ 2N × 2N : Tη,ηRTξ,ξ =∗ λTη,ξ, λ ∈ U}

is Borel in 2N × 2N. In particular, if BRU is either countable or of size of the

continuum.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 7 / 8

Example 1For each ξ ∈ 2N we can associate a set Aξ = {s ∈ 2<N : s ⊆ ξ}.

FactLet X ⊆ N. Then for each λ ∈ {0,1} the sets {ξ ∈ 2N : Aξ ∩ X =∗ λAξ} areBorel.

For each pair (ξ, η) ∈ 2N × 2N we associate an operator on `2(2<N)

Tη,ξ(s) =

{eη|k if s = eξ|k for some k ∈ N0 otherwise

Lemma

Let R ∈ B(H) and U be a Borel subset of C, then the set

BRU = {(η, ξ) ∈ 2N × 2N : Tη,ηRTξ,ξ =∗ λTη,ξ, λ ∈ U}

is Borel in 2N × 2N. In particular, if BRU is either countable or of size of the

continuum.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 7 / 8

Example 1For each ξ ∈ 2N we can associate a set Aξ = {s ∈ 2<N : s ⊆ ξ}.

FactLet X ⊆ N. Then for each λ ∈ {0,1} the sets {ξ ∈ 2N : Aξ ∩ X =∗ λAξ} areBorel.

For each pair (ξ, η) ∈ 2N × 2N we associate an operator on `2(2<N)

Tη,ξ(s) =

{eη|k if s = eξ|k for some k ∈ N0 otherwise

Lemma

Let R ∈ B(H) and U be a Borel subset of C, then the set

BRU = {(η, ξ) ∈ 2N × 2N : Tη,ηRTξ,ξ =∗ λTη,ξ, λ ∈ U}

is Borel in 2N × 2N. In particular, if BRU is either countable or of size of the

continuum.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 7 / 8

Example 1For each ξ ∈ 2N we can associate a set Aξ = {s ∈ 2<N : s ⊆ ξ}.

FactLet X ⊆ N. Then for each λ ∈ {0,1} the sets {ξ ∈ 2N : Aξ ∩ X =∗ λAξ} areBorel.

For each pair (ξ, η) ∈ 2N × 2N we associate an operator on `2(2<N)

Tη,ξ(s) =

{eη|k if s = eξ|k for some k ∈ N0 otherwise

Lemma

Let R ∈ B(H) and U be a Borel subset of C, then the set

BRU = {(η, ξ) ∈ 2N × 2N : Tη,ηRTξ,ξ =∗ λTη,ξ, λ ∈ U}

is Borel in 2N × 2N. In particular, if BRU is either countable or of size of the

continuum.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 7 / 8

Example 1For each ξ ∈ 2N we can associate a set Aξ = {s ∈ 2<N : s ⊆ ξ}.

FactLet X ⊆ N. Then for each λ ∈ {0,1} the sets {ξ ∈ 2N : Aξ ∩ X =∗ λAξ} areBorel.

For each pair (ξ, η) ∈ 2N × 2N we associate an operator on `2(2<N)

Tη,ξ(s) =

{eη|k if s = eξ|k for some k ∈ N0 otherwise

Lemma

Let R ∈ B(H) and U be a Borel subset of C, then the set

BRU = {(η, ξ) ∈ 2N × 2N : Tη,ηRTξ,ξ =∗ λTη,ξ, λ ∈ U}

is Borel in 2N × 2N. In particular, if BRU is either countable or of size of the

continuum.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 7 / 8

Example 2

FactLet A be MAD and X ⊆ N infinite. Then

{A ∩ X : A ∈ A}

is MAD in ℘(X ).

If P,Q are projections, then PQ is not a projection unless P and Q commute.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 8 / 8

Example 2

FactLet A be MAD and X ⊆ N infinite. Then

{A ∩ X : A ∈ A}

is MAD in ℘(X ).

If P,Q are projections, then PQ is not a projection unless P and Q commute.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 8 / 8

Example 2

FactLet A be MAD and X ⊆ N infinite. Then

{A ∩ X : A ∈ A}

is MAD in ℘(X ).

If P,Q are projections, then PQ is not a projection unless P and Q commute.

Piotr Koszmider (IM PAN) A non-commutative Mrowka’s Ψ-space Oaxaca, 13-09-2016 8 / 8